Common Core Learning Standard
Students in grade 6 build on their work with area in elementary school by reasoning about relationships among shapes to determine area, surface area, and volume. They will find areas of right triangles, other triangles, and special quadrilaterals by decomposing these shapes, rearranging or removing pieces, and relating the shapes to rectangles. Using these models, students will discuss, develop, and justify formulas for areas of triangles and parallelograms. Students will find areas of polygons and surface areas of prisms and pyramids by decomposing them into pieces whose area they can determine. They will reason about right rectangular prisms with fractional side lengths to extend formulas for the volume of a right rectangular prism to fractional side lengths. They will prepare for work on scale drawings and constructions in grade 7 by drawing polygons in the coordinate plane. The Mathematical Practices should be evident throughout instruction and connected to the content addressed in this unit. Students should engage in mathematical tasks that provide an opportunity to connect content and practices.
Big Ideas
•Students will demonstrate understanding of how to calculate area of two-dimensional figures through the use of mathematical formulas and use of geometric nets.
•Students will demonstrate understanding of how to calculate volume and surface area of three-dimensional figures through the use of mathematical formulas.
Essential Concepts
• The area of irregular and regular polygons can be found by decomposing the polygon into rectangles, triangles and other shapes.
• Manipulatives and the construction of nets may be used in computing the surface area of rectangular and triangular prisms, and volume of right rectangular prism.
• Formulas may be used to compute the areas of polygons, surface areas of rectangular and triangular prisms, and volumes of right rectangular prisms.
• Appropriate units of measure should be used when computing the area (square units) of polygons, and surface area (square units) and volume of prisms (cubic units).
• Views of rectangular and triangular prisms may be interpreted and sketched to provide a 2-dimensional representation of a three dimensional figure.
• Fractional edge lengths are equivalent to the dimensions of solid figures
• The volume of a solid figure is the number of same sized cubes filling the space so that there are no gaps and overlaps.
Vocabulary
• 2-Dimensional: A shape that only has two dimensions (such as width and height) and no thickness.
• 3-Dimensional: An object that has height, width and depth (thickness), like any object in the real world.
• Area: The number of square units it takes to completely fill a space or surface.
• Bases of a Prism: The two faces formed by congruent polygons that lie in parallel planes, all of the other faces being parallelograms.
• Cubic Units: Volume of the solids is measured in Cubic Units.
• Edge: The intersection of a pair of faces in a three-dimensional figure.
• Equilateral Triangle: A triangle which has all three of its sides equal in length.
• Face: One of the polygons that makes up a polyhedron.
• Fractional edge length: The length of each edge of the cube is a fraction.
• Isosceles Triangle: A triangle which has two of its sides equal in length.
• Kite: A quadrilateral with two distinct pairs of equal adjacent sides. A kite-shaped figure.
• Lateral Faces: In a prism, a face that is not a base of the figure.
• Net: A two-dimensional figure that, when folded, forms the surfaces of a three-dimensional object.
• Parallelogram: A quadrilateral with both pairs of opposite sides parallel.
• Polygon: A number of coplanar line segments, each connected end to end to form a closed shape. A regular polygon has all sides equal and all interior angles equal. An irregular polygon sides are not all the same length nor does the interior angles have the same measure.
• Polyhedron: A 3-dimensional figure that has polygons as faces.
• Prism: A polyhedron with two parallel and congruent faces, called bases, and all other faces that are parallelograms.
• Quadrilaterals: Four coplanar line segments linked end to end to create a closed figure. A 4-sided polygon.
• Rectangle: A 4-sided polygon where all interior angles are 90°.
• Rectangular prism: A solid (3-dimensional) object which has six faces that are rectangles.
• Rhombus: A quadrilateral with all four sides equal in length.
• Right Triangle: A triangle where one of its interior angles is a right angle (90 degrees).
• Right rectangular prism: In a right prism, the lateral faces are each perpendicular to the bases.
• Scalene Triangle: A triangle where all three sides are different in length.
• Square: A quadrilateral that has four right angles and four equal sides.
• Surface area: The total area of the 2-dimensional surfaces that make up a 3-dimensional object.
• Trapezoid: A quadrilateral which has one pair of parallel sides.
• Triangles: A closed figure consisting of three line segments linked end-to-end. A 3-sided polygon
• Triangular prism: A prism whose bases are triangles. A solid (3-dimensional object what has five faces: three rectangles and two bases.
• Vertices: The common endpoint of two or more rays or line segments
• Volume: The amount of space occupied by an object.
• Volume of a Prism: The area of a base times the height. The number of cubic units to fill a prism.
Content
In this unit students will:
• Find areas of right, equilateral, isosceles, and scalene triangles, and special quadrilaterals
• Find areas of composite figures and polygons by composing into rectangles and decomposing into triangles and other shapes
• Solve real-world and mathematical problems involving area
• Decipher and draw views of rectangular and triangular prisms from a variety of perspectives
• Recognize and construct nets for rectangular and triangular prism
• Find the surface area of rectangular and triangular prisms by using manipulatives and by constructing nets
• Determine the surface area of rectangular and triangular prisms by substituting given values for their dimensions into the correct formulas;
• Solve real-world that require determining the surface area of rectangular and triangular prisms
• Measure and compute volume with fractional edge length using cubic units of measure
• Find the volumes of right rectangular prisms by substituting given values for their dimensions into the correct formulas
• Make the connection that finding the volume given the length (l) x width (w) is the same as the base (B)
• Solve real-world problems that require determining the volume of right rectangular prism
Skills Needed
•number sense
• computation with whole numbers and decimals, including application of order of operations
• multiplication and division of fractions
• formulas for finding area, surface area and volume
• area measures in square units and volume measures in cubic units
• properties of polygons, 2-D, and 3-D shapes
Solve real-world and mathematical problems involving area, surface area, and volume.
6.G.1
Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.
6.G.3
Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems.
6.G.4
Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems.
Bridge Guidance
5.MD.3
Recognize volume as an attribute of solid figures and understand concepts of volume measurement.
a. A cube with side length 1 unit, called a “unit cube,” is said to have “one cubic unit” of volume, and can be used to measure volume.
b. A solid figure which can be packed without gaps or overlaps using n unit cubes is said to have a volume of n cubic units.
5.MD.4
Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units.
5.MD.5
Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume.
a. Find the volume of a right rectangular prism with whole-number side lengths by packing it with unit cubes, and show that the volume is the same as would be found by multiplying the edge lengths, equivalently by multiplying the height by the area of the base. Represent threefold whole-number products as volumes, e.g., to represent the associative property of multiplication.
b. Apply the formulas V=l×w×h and V=b×h for rectangular prisms to find volumes of right rectangular prisms with whole- number edge lengths in the context of solving real world and mathematical problems.
c. Recognize volume as additive. Find volumes of solid figures composed of two non-overlapping right rectangular prisms by adding the volumes of the non-overlapping parts, applying this technique to solve real world problems.
Table of Contents
Common Core Learning Standard
Students in grade 6 build on their work with area in elementary school by reasoning about relationships among shapes to determine area, surface area, and volume. They will find areas of right triangles, other triangles, and special quadrilaterals by decomposing these shapes, rearranging or removing pieces, and relating the shapes to rectangles. Using these models, students will discuss, develop, and justify formulas for areas of triangles and parallelograms. Students will find areas of polygons and surface areas of prisms and pyramids by decomposing them into pieces whose area they can determine. They will reason about right rectangular prisms with fractional side lengths to extend formulas for the volume of a right rectangular prism to fractional side lengths. They will prepare for work on scale drawings and constructions in grade 7 by drawing polygons in the coordinate plane. The Mathematical Practices should be evident throughout instruction and connected to the content addressed in this unit. Students should engage in mathematical tasks that provide an opportunity to connect content and practices.
Big Ideas
•Students will demonstrate understanding of how to calculate area of two-dimensional figures through the use of mathematical formulas and use of geometric nets.
•Students will demonstrate understanding of how to calculate volume and surface area of three-dimensional figures through the use of mathematical formulas.
Essential Concepts
• The area of irregular and regular polygons can be found by decomposing the polygon into rectangles, triangles and other shapes.
• Manipulatives and the construction of nets may be used in computing the surface area of rectangular and triangular prisms, and volume of right rectangular prism.
• Formulas may be used to compute the areas of polygons, surface areas of rectangular and triangular prisms, and volumes of right rectangular prisms.
• Appropriate units of measure should be used when computing the area (square units) of polygons, and surface area (square units) and volume of prisms (cubic units).
• Views of rectangular and triangular prisms may be interpreted and sketched to provide a 2-dimensional representation of a three dimensional figure.
• Fractional edge lengths are equivalent to the dimensions of solid figures
• The volume of a solid figure is the number of same sized cubes filling the space so that there are no gaps and overlaps.
Vocabulary
• 2-Dimensional: A shape that only has two dimensions (such as width and height) and no thickness.
• 3-Dimensional: An object that has height, width and depth (thickness), like any object in the real world.
• Area: The number of square units it takes to completely fill a space or surface.
• Bases of a Prism: The two faces formed by congruent polygons that lie in parallel planes, all of the other faces being parallelograms.
• Cubic Units: Volume of the solids is measured in Cubic Units.
• Edge: The intersection of a pair of faces in a three-dimensional figure.
• Equilateral Triangle: A triangle which has all three of its sides equal in length.
• Face: One of the polygons that makes up a polyhedron.
• Fractional edge length: The length of each edge of the cube is a fraction.
• Isosceles Triangle: A triangle which has two of its sides equal in length.
• Kite: A quadrilateral with two distinct pairs of equal adjacent sides. A kite-shaped figure.
• Lateral Faces: In a prism, a face that is not a base of the figure.
• Net: A two-dimensional figure that, when folded, forms the surfaces of a three-dimensional object.
• Parallelogram: A quadrilateral with both pairs of opposite sides parallel.
• Polygon: A number of coplanar line segments, each connected end to end to form a closed shape. A regular polygon has all sides equal and all interior angles equal. An irregular polygon sides are not all the same length nor does the interior angles have the same measure.
• Polyhedron: A 3-dimensional figure that has polygons as faces.
• Prism: A polyhedron with two parallel and congruent faces, called bases, and all other faces that are parallelograms.
• Quadrilaterals: Four coplanar line segments linked end to end to create a closed figure. A 4-sided polygon.
• Rectangle: A 4-sided polygon where all interior angles are 90°.
• Rectangular prism: A solid (3-dimensional) object which has six faces that are rectangles.
• Rhombus: A quadrilateral with all four sides equal in length.
• Right Triangle: A triangle where one of its interior angles is a right angle (90 degrees).
• Right rectangular prism: In a right prism, the lateral faces are each perpendicular to the bases.
• Scalene Triangle: A triangle where all three sides are different in length.
• Square: A quadrilateral that has four right angles and four equal sides.
• Surface area: The total area of the 2-dimensional surfaces that make up a 3-dimensional object.
• Trapezoid: A quadrilateral which has one pair of parallel sides.
• Triangles: A closed figure consisting of three line segments linked end-to-end. A 3-sided polygon
• Triangular prism: A prism whose bases are triangles. A solid (3-dimensional object what has five faces: three rectangles and two bases.
• Vertices: The common endpoint of two or more rays or line segments
• Volume: The amount of space occupied by an object.
• Volume of a Prism: The area of a base times the height. The number of cubic units to fill a prism.
Content
In this unit students will:
• Find areas of right, equilateral, isosceles, and scalene triangles, and special quadrilaterals
• Find areas of composite figures and polygons by composing into rectangles and decomposing into triangles and other shapes
• Solve real-world and mathematical problems involving area
• Decipher and draw views of rectangular and triangular prisms from a variety of perspectives
• Recognize and construct nets for rectangular and triangular prism
• Find the surface area of rectangular and triangular prisms by using manipulatives and by constructing nets
• Determine the surface area of rectangular and triangular prisms by substituting given values for their dimensions into the correct formulas;
• Solve real-world that require determining the surface area of rectangular and triangular prisms
• Measure and compute volume with fractional edge length using cubic units of measure
• Find the volumes of right rectangular prisms by substituting given values for their dimensions into the correct formulas
• Make the connection that finding the volume given the length (l) x width (w) is the same as the base (B)
• Solve real-world problems that require determining the volume of right rectangular prism
Skills Needed
•number sense
• computation with whole numbers and decimals, including application of order of operations
• multiplication and division of fractions
• formulas for finding area, surface area and volume
• area measures in square units and volume measures in cubic units
• properties of polygons, 2-D, and 3-D shapes
on
Topic
Lessons
Pages
Resources
Arc 1 - 2D
real world application
Activity Lab 9-4a
425-433 #1-10,15-20,25-31,35,36
real world application
Activity Lab 9-4a
425-433 #1-10,15-20,25-31,35,36
real world application
Activity Lab 9-4a
425-433 #1-10,15-20,25-31,35,36
520-521
Arc 2 - 3D
real world application of skill
235-238
434-448
real world application of skill
235-238
434-448
Unit 6 Assessments
a. A cube with side length 1 unit, called a “unit cube,” is said to have “one cubic unit” of volume, and can be used to measure volume.
b. A solid figure which can be packed without gaps or overlaps using n unit cubes is said to have a volume of n cubic units.
a. Find the volume of a right rectangular prism with whole-number side lengths by packing it with unit cubes, and show that the volume is the same as would be found by multiplying the edge lengths, equivalently by multiplying the height by the area of the base. Represent threefold whole-number products as volumes, e.g., to represent the associative property of multiplication.
b. Apply the formulas V=l×w×h and V=b×h for rectangular prisms to find volumes of right rectangular prisms with whole- number edge lengths in the context of solving real world and mathematical problems.
c. Recognize volume as additive. Find volumes of solid figures composed of two non-overlapping right rectangular prisms by adding the volumes of the non-overlapping parts, applying this technique to solve real world problems.